Nuclear magnetism and the electron state in low ... · interacting electrons hyperfine interaction A ~ 90 μeV (GaAs) A ~ 0.6 μeV (13C) dipolar/quadrupolar interactions < 0.1 neV

Nuclear magnetism and the electron state in low-dimensionalconductors

Bernd BrauneckerUniversidad Autónoma de Madrid, Spain

in collaboration with (mainly)

Daniel Loss (Basel)Pascal Simon (Orsay)

Motivation: Why nuclear magnetism?

Electron spin trappedin quantum dot

Main culprit: Ensemble of ~ 105 nuclear spins within envelope of electronwave function

Well-defined quantum state may be used as a qubit.

Problem: Decoherence due to interaction withenvironment

B. Braunecker, Varenna 2012


Decoherence substantiallyslowed down if nuclearspins are fully polarized

How to achieve polarization?

through coupling to electrons

electrons should do it themselves



Decoherence substantiallyslowed down if nuclearspins are fully polarized

How to achieve polarization?

through coupling to electrons

electrons should do it themselves

required: many electrons



look atnuclear spins in a conductor



Can we obtain nuclear magnetic order

intrinsically through a phase transition?


3D metals: nuclear ferromagnet; old story Weiss mean-field theory (Fröhlich & Nabarro, 1940)

Dimensionality matters: interactions become important through restriction of scattering phase space

3D

2D

1D

2D: RKKY interaction renormalized through electron-electron interactions can stabilize nuclear magnetic order

1D: Renormalization is essential; electrons and nuclear spins can form a combined ordered state of matter

Can we obtain nuclear magnetic order

intrinsically through a phase transition?


2D

Order possible only due to non-Fermi liquid corrections by electron-electron interactions.

Estimated transition temperatures up to 10 – 100 μK for strong interactions.

Nuclear ferromagnet

Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008


Novel collective phase: combined state of

nuclear spin helixhelical electron spin density wave helical conductor

Transition temperatures up to 100 mK possible in GaAs wires.

1D

Ferromagnetic locking on cross-section

2kF spin rotation along wire

Nuclear helimagnet

relevantbackactionbetween electrons andnuclear spins

schematic wire

Braunecker, Simon, Loss PRL 2009, PRB 2009


Novel collective phase: combined state of

nuclear spin helixhelical electron spin density wave helical conductor

Transition temperatures up to 100 mK possible in GaAs wires

1D

Ferromagnetic locking on cross-section

2kF spin rotation along wire

Nuclear helimagnet

relevantbackactionbetween electrons andnuclear spins

a bit more realistic wire



Microscopic model

Microscopic model

interactingelectrons

hyperfine interaction

A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)

dipolar/quadrupolar interactions< 0.1 neV (smallest energy scales)

A priori a tremendously complicated mixture of 3D nuclear spinsand low-dimensional electrons.

→ reduce to treatable effective model


Reduction to an effective model

Reduction to effective model (I)



A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)


electron spin nuclear spin


Reduction to effective model (II)

Focus only on the nuclear spins within the support of the confined electron liquid.

unoccupiedprojection ontolowest subband nuclear spins in

transverse direction (cross-section)

with

projectedelectron spin

compositenuclear spinof length I

The hyperfine interaction reduces to the coupling between projected electron spins and composite nuclear spins

Expand electron operators in basis of transverse modes t0, t1, ...

If there is order, energy is minimized if all spins in the composite spin are parallel: ferromagnetic on the cross-section


Reduction to effective model (III)



A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)


small number compared with EF ~ 10 – 100 meV

separation of time scales

• electrons move in static nuclear background (Overhauser field)• nuclear spins see an instantaneously reacting electron gas→ 2 coupled effective models

(similar to Born-Oppenheimer approx.)

electron spin nuclear spin


Reduction to effective model (III): RKKY

Schrieffer-Wolff transformation;integrate out electron degrees of freedom

static electron spin susceptibility

RKKY interaction Jij = J(ri-rj)long ranged

A / EF ~ 1/100 (or smaller): separation of time scales between electrons & nuclear spins


Effective model

effective electron Hamiltonian

effective nuclear spin Hamiltonian

electrons in staticnuclear background(Overhauser field)

nuclear spins with RKKY interaction transmittedby electrons


Effective model

effective electron Hamiltonian

effective nuclear spin Hamiltonian

mutualdependence

novel physics

self-consistency


Mean-field analysis(naive but instructive)

Mean-field theory

similar to Fröhlich & Nabarro (1940)

Ground state determined by minimum of Jq

if at q = 0: ferromagnetif at q ≠ 0: helimagnet

Theory depends on single energy scale: TMF = |min(Jq)|, e.g. J0


Mean-field theory

similar to Fröhlich & Nabarro (1940)

If a system is characterizedby a single energy scale, thismean-field argument is valid

Thermodynamics controlled by single scale TMF

e.g., Curie temperaturefor ordering transition:


Order in : role of interactions2D

Order in 2D: role of interactions

susceptibility at T=0

Noninteracting electrons:Lindhard functionno unique minimum: no nuclear spin order possible




Interacting electrons:Non-Fermi-liquid corrections of self-energy lead to nonanalytic contribution, linear in |q|.

dominant self-energy renormalization of Cooper channel scattering amplitude

Chubukov, Maslov PRB 2003Aleiner, Efetov PRB 2006Saraga, Altshuler, Loss, Westervelt PRB 2005Shekhter, Finkel'stein PRB 06, PNAS 2006


Simon, Braunecker, Loss PRB 2008Chesi, Żak, Simon, Loss PRB 2009Żak, Maslov, Loss PRB 2010, 2012




Interacting electrons:Non-Fermi-liquid corrections of self-energy lead to nonanalytic contribution, linear in |q|.


However, the sign of the linear correction seems to be nonuniversal.

Possible other outcomes:



Consequences

minimum at q = 0: nuclear ferromagnet MF scale: TMF ~ J0

interaction-induced scale

2 energy scales: pure MF theory not applicable

Which physics is described by this susceptibility?



Fluctuations

MF scale: TMF ~ J0

interaction-induced scale

Spin-wave (magnon) analysis of fluctuations

T0 >> T* , independent of TMF

modified Bloch law

Calculate magnetization m per nuclear spin at T < T*

T0 provides an estimate of Tc, reaching up to the mK range - consistent with noninteracting limit: T* = 0 → T0 = 0 - TMF absent (but role not clear)

finite because ωq

is linear at small q

T0 ~ 1 mK for rs ~ 10

main message:m > 0 for 0 < T < T*

magnon spectrum: ωq = |J0 – Jq | ~ |q|

B. Braunecker, Varenna 2012 Simon, Loss PRL 2007; Simon, Braunecker, Loss PRB 2008

Generalized Mermin-Wagner theorem

Order in 2D: What about the Mermin-Wagner Theorem?

“No long-range order in 2D for sufficiently short-ranged interactions.”

● RKKY interactions are long ranged

● Is there some extension of the theorem?

Yes: General proof of absence of long-range order for"a wide class of models including any form of electron-electron and single-electron interactions that are independent of spin"

Loss, Pedrocchi, Leggett PRL 2011

Mermin, Wagner PRL 1966


Generalized Mermin-Wagner theorem

Are we in trouble? — No!

1) Nonequilibrium situation:T=0 for electrons andT>0 for nuclear spins

2) Finite system size in μm rangeprovides cutoff (thermal lengthis larger)→ practically most important

3) Backaction of the nuclear spinscreates a small excitation gap→ fundamentally important


Backaction on electronsself-consistency

The ordered nuclear spins generate the Overhauser field, similar to a magnetic field

→ small electron polarization→ makes RKKY interaction spin-dependent and anisotropic→ does no longer fulfil the conditions of the theorem→ creates small excitation gap for the magnons

A very similar effect is obtained by spin-orbit interactions in the 2DEG.Żak, Maslov, Loss PRB 2012

True long-range order could actually be possible.

Simon, Braunecker,Loss PRB 2008


Summary of 2D

A nuclear ferromagnet in a 2DEG is possible and stable

– crucial is the non-Fermi liquid modification of the RKKY interaction by electron-electron interactions

– the self-consistent backaction between electrons and nuclear spins provides conditions that would in principle even allow for true long-range order

– estimates for transition temperatures are in the range 10 – 100 μK reaching for very strong interactions into the mK range



Order in : spectacular results from strong non-Fermi liquid physics

1D

Electrons in 1D

Interacting electron system expressed by bosonic field theory

charge/spindensity fluctuations

conjugated fields (→ currents)

electron-electron interactions:strong renormalization of - velocities: compressibility/susceptibility- charge/spin fraction of density waves

electron Coulomb interaction

another electron

Model interacting electrons by Luttinger liquid


RKKY interaction from the Luttinger liquid

electron / nuclear spin density

thermal lengthinteraction-dependentexponent

for a Luttinger liquidthe electron spin susceptibilitycan be calculated explicitly

→ strongly susceptible at 2kF

→ determines RKKY interaction

cf. also Egger and Schoeller, PRB (1996)

B. Braunecker, Varenna 2012 Braunecker, Simon, Loss PRL 2009, PRB 2009

Nuclear spin ground state: Solve by inspection...

nuclear helimagnet

● minimize energy:all weight in Fourier modesq = ± 2kF

● ferromagnetic locking on cross-section: large spinof maximal length

stability?

minimum at ±2kF


confirmed by magnon calculation;finite size cutoff is required (no long-range order),but transition temperature is independent of size

Stability of the helical order

thermal lengthdepth

depth & width determined by T

theory depends on a single scale:the mean-field argument applies

sets crossover temperature

width



Nuclear helimagnet

Helical order of nuclear spins helimagnet along the wire

Order stable up to temperature T*

magnetization

generates spiral Overhauser fieldeffect on electrons?



Electrons are susceptible at q = 2kF

Nuclear magnetic (Overhauser) field

spatial frequency q = 2kF

at which electrons are extremely susceptible

Overhauser field drives instability that exists in any 1D system

→ Peierls transition

self-consistency



The electronic Peierls transition

one-dimensional conductor with external periodic potentialof spatial frequency 2kF

scattering between Fermi pointsopens gap

system becomes insulating


The spin-selective Peierls transition

external periodic potential:spiral magnetic fieldof spatial frequency 2kF

spin-selective scattering between Fermi pointsopens gap for 1/2 of the modes

1/2 of the system becomes insulating and forms a spiral density wave1/2 of the system remains conducting and forms a helical conductor

Δ


Braunecker, Klinovaja, Japaridze,Loss PRB 2010

Enhancement by electron-electron interactions

Feedback of Overhauser field on electrons

Bosonization treatment: relevant sine-Gordon interaction

Gap for

strongly enhanced

spiral electron spindensity wave;combines spinand charge fields


Flow to strong coupling

effective Overhauserfield flows to strongcoupling limit

ξ : correlation lengthprecise form depends on material

renormalization absent in Fermi liquids (Kc = Ks = 1)

coupling constantRG equation for dimensionless



Changed system properties

1) enhanced gap Δ* µ A* for mode

2) remaining gapless mode

• renormalized Luttinger liquid• spin-filtered; helical conductor• RKKY interaction maintains its

singular shape at 2kF

• Jq becomes deeper

R ↑

L ↓

Δ*



Renormalized RKKY interaction

- same shape- much deeper (modified exponents)- boosts T*

RKKY interaction determined by remaining gapless modes

modified exponents / prefactors



Renormalized RKKY interaction

- same shape- much deeper (modified exponents)- boosts T*

RKKY interaction determined by remaining gapless modes

T* still lower than renormalized A*: order vanishes by melting of nuclear spin alignment

T* = 10 – 100 mK GaAs quantum wires (depending on interaction strength)

T* ~ 10 mK carbon nanotubes



Combined electron / nuclear spin order

pinned electron density wave (electron spin helix)

nuclear helimagnet

Below T* the ordered phases depend on each other:

- T* ~ 10 – 100 mK - huge renormalization due to electron-electron interactions- gapless modes: helical conductor

Phase of tightly bound electron & nuclear spin degrees of freedom


Further consequences

Reduced conductance, helical conductor

Through backaction: Pinning of channels

Blocking of ½ of the conducting channels

Universal reduction of conductance by factor 2

Remaining channels are helical

R ↑

L ↓

spin-filter

s-wave superconductor

Majorana bound states

Still requires a complete self-consistent stability analysis!


Anisotropy in electron spin susceptibility

Overhauser field defines spin (x,y) plane

Anisotropy between spin (x,y) and z directions



Unusual "irregular" density of states

Braunecker, Bena, Simon PRB 2012Schuricht PRB 2012

Local (tunneling) density of states has an "irregular" contributionarising from coupling to the gapped sector

Perfect helical LL: regular LL behavior

Helical LL in combined ordered phase(spiral LL): regular LL behavior + irregular contribution (pseudogap)

Possible to use local DOS to prove existence of nuclear orderor of helical states in nanowires

regular behaviorbelow gap

irregularbehaviordominatesgap edge

irregular exponentchanges signfor strong interactions


Summary of 1D

A nuclear helimagnet in a 1D wire is possible and stable

– crucial is the backaction between electrons and nuclear spins: it restructures the electron state and the resulting order consists of

– nuclear spin helix

– helical electron spin density wave

– helical electron conductor

– electron interactions strongly renormalize and stabilize the state

and transition temperatures up to 100 mK in GaAs are possible



Global Conclusions

3D

2D

1D

2D: RKKY interaction renormalized through electron-electron interactions cana) overrule generalized Mermin-Wagner Theoremb) stabilize nuclear magnetic order

1D: Renormalization is essential; electrons and nuclear spins can form a combined state of matter

Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008Braunecker, Simon, Loss PRL 2009, PRB 2009


Documents

Nuclear magnetism and the electron state in low ... · interacting electrons hyperfine interaction A ~ 90 μeV (GaAs) A ~ 0.6 μeV (13C) dipolar/quadrupolar interactions < 0.1 neV